Ok...
Re: Calculation of Risk of Loss
Ok...
1) The probility distribution for a single hand of video poker is
NOT NORMAL. Moreover, it
is discrete (non-continuous) function that does not approximate a
sampled (descritized)
normal distribution. The PDF is not symetric about the mean or
mode. Indeed, its mean
(simple arithmetric average) is always larget than the mode (most
likely value, or peak)
becuase of the heavy positve tail (RF's!).
Agree
2) The result of a "session" is the sum of the outcomes of the
hands that make up the
session. Given the PDF of a single hand-- which is known perfectly
(assuming you play a
fixed strategy, etc) and the number of hands, and any constraing on
bankroll-- one can
compute the PDF for a session (consisting of said number of
hands). The central limit
theorm holds for this PDF (since its values are assembled from the
SUM of indiviudal hands
that have FINITE, though skewed PDF's themselves). The CLT does
not say that the session
PDF becomes normal eveywhere. Instead, it says that in
the "central reegion", the PDF
becomes normal. This central region also looks symettric, with the
mean and the mode all
lining up (almost) as they should for a normal distribution.
Outside the "central region",
the PDF in NOT NORMAL. As the number of hands increases,
the "area" of the central
region compared to the "tails" increases-- and one might as well
say that the whole thing
is normal.
NOTE: this stuff is all about the computed
distribution for a session.
Not sure what you mean by that last sentence
I would have thoght the distribution of hands would be very skewed
with a high concentration around the lower paying hands (I further
think the highest concentration would be around no payback at all)
and way out on the right side would be the royal.
3) If you take your session results and assemble a PDF, you may or
may not get something
that looks like a desritized version of a normal distribution.
Most likely, your results will
sample the meat of the PDF-- that is you won't get a really really
big win or loss. So your
curve would look more normal-like that it really is-- simple
because you don't have
enough data-- nor could you ever get enough data.
If I define a session as 1000 hands, seems like I could get quite a
bit of data over a fairly short period of time.
4) To understand how non-normal random numbers when ADDED up turn
into something
normal, you can do some easy experiments.... You can do this on
your PC or even with a
dice. BAscially generate a set of random numbers with an even
distribution (the kind of
number you get from a single die). Note the mean and variance of
the set of humbers.
Then create a new set of numbers by adding a certain number of the
original random
numbers. For example, you generate a list of 1000 random
numbers... call them X's. Now
generate a list of Y's by adding together four X's to create 1 Y (
you can average them also
by dividing by 4 if you want). Now compute the mean an variance
again. If you divided
them all by 4, you will find that the mean stayed the same, but the
variance SHRUNK!
Moreover, if you plot the distribution, you would see it is startin
gto look normal! Woo-
hoo. BTW, you can do this experiment as a function of the number
of X's you add up.
While the orihinal x's have a flat-ebven distribution, as you add
them up, you get a
triangle distribution, then something more normal, and ultimately,
a single peak!
BTW, there are a lot of demos on the web of this. Here is one for
dice. http://
www.stat.sc.edu/~west/javahtml/CLT.html
In this case, 1 dice = X's
2 dice = Y = x +x
3 dice = Y = x +x +x
They didn't divide by the number of dice (x's) so the mean moves...
but the effect is the
same.
I feel that I understand the CLT, but plotting the outcomes of a
session, to me is a plot of individuals and not a plot of samples be
they sums, differences, averages etc of the parent distribution.
I agree if I were sampling by combining/adding/averaging (for
example every 25 hands) of the individual outcomes, then the
distribution would start to look more normal. However, I repeat it
seems to me a distribution of a session (whether the session is 1000
hands or 1 million hands) would represent 1000 or 1 million or what
ever defines a session data points and thus would look like the
expected probabilities per the various outcome, which would
decidedly be non-normal.
···
--- In vpFREE@yahoogroups.com, "cdfsrule" <vpfree_digests@...> wrote:
--- In vpFREE@yahoogroups.com, "cdfsrule" <vpfree_digests@...> wrote:
